As noted in the comments, by the spherical symmetry, the distribution of the dot product in both parts of your question is the same that of $X\cdot(1,0,\dots,0)$. Moreover, the distribution of $X$ is the same as that of the random vector
$$\frac{Z}{\sqrt{Z_1^2+\dots+Z_d^2}},$$
where $Z=(Z_1,\dots,Z_d)$ is a standard normal random vector.
So, the distribution of the dot product in question is the same that of
$$R:=\frac{Z_1}{\sqrt{Z_1^2+\dots+Z_d^2}}.$$
The distribution of $R$ is obviously symmetric, and the distribution of $R^2$ is the beta distribution with parameters $\frac12,\frac{d-1}2$. It follows that the probability density function (pdf) $f_R$ of $R$ is given by
$$f_R(r)=\frac{\Gamma \left(\frac{d}{2}\right)}{\sqrt{\pi }\, \Gamma
\left(\frac{d-1}{2}\right)}\,\left(1-r^2\right)^{\frac{d-3}{2}}\, 1\{|r|<1\},$$
and the dot product in question has the same pdf.